I wired in a Trigorilla mosfet like the modified wiring schematic shows, but now PSU only gives off a high pitched buzzing when the printer is switched on. (It sounds like it's coming from the transformer in the middle of the board.) I can unhook the two wires that run power between the PSU and controller board and the PSU will then work like normal, but of course the controller is off then. When I opened up the PSU, the NTC thermister had failed and was sparking. I replaced it, but that still doesn't solve the issue of the not being able to connect power to the controller board. Does anyone have an idea of what the problem might be?

Yes, that is correct. I had the connector for the print bed on the controller board melt, so I had to replace that, and then I added the MOSFET with the hopes that it will prevent a occurrence. Now, as long as the controller is wired to the PSU, the PSU won't run fans or anything else and only makes a buzzing sound. I tested the replaced heat bed connector on the controller (which now ties into the MOSFET) as the soldering wasn't perfect, but it didn't have a short.

I tested both sides of the connector with a voltmeter and there wasn't any connection between them

I assume you are testing for ohms. I was more concerned that a solder bridge may have been created between one of the pins and a neighbouring connection or trace. That would not necessarily show when testing continuity between the pins.

It is starting to sound like the controller. I would try removing all connections from it. Then try connecting it to just a computer via USB, (also remove the steppers) and see if it boots? If the the controller does not work with just the USB or PSU, then it is time for more serious debugging on the controller card, or get a new one. If it works with the USB, but not the PSU, then there may be a problem with the PSU. Try putting a load on it with something other then the controller. You can try the heated bed, but don't leave it on long (more than a few minutes) as there is no control, and it will quickly become a hazard.

I plugged the controller into my computer and the LEDs lit up. It wouldn't connect in Pronterface at first and gave me: [ERROR] Could not connect to COM3 at baudrate 250000:Serial error: could not open port COM3: [Error 2] The system cannot find the file specified.I fixed this by reuploading Marlin, after which it connected.

When the PSU was turned on and connected, the PSU continued having the same issue as before. I unhooked it from the controller and it started to run normally as before. I plugged the heat bed directly into the PSU and it started to heat up, so it seems to function, as does the controller.

Well you are definitely narrowing down the issue, good work. If nothing else is connected to the controller when testing with the PSU, then something is shorting on the 12v side of the controller. There is a small switching regulator on the controller that takes the 12v and provides the 5v needed to power much of the electronics. This is most likely isolating the 5v from the USB and the issue (short) on the 12v side of the controller's electronics. This will require a closer look at the controller. Were there any other issues or symptoms when the connector overheating occurred?

I noticed the connector overheating because the bed would no longer heat up properly. I opened up the electronics and checked the temperature with a heat gun, which indicated that the wires going from the controller to the bed were really hot. This led to the connector which had started to melt and deform. I then looked it up and found that this was a fairly common issue. I added the MOSFET which was what people recommended to resolve the problem. Nothing else seemed off though, besides the melted connector, until I tried to start up the printer after fixing and rewiring things.

Well you may have to start looking at the boards circuitry and debugging down to the component level. I am not sure I would bother as a replacement is only $30. Something either failed as a short or your modification has introduced an issue. Are you seeing a dead short (0 ohms) when checking between the power inputs?